Step |
Hyp |
Ref |
Expression |
1 |
|
lcvnbtwn.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lcvnbtwn.c |
⊢ 𝐶 = ( ⋖L ‘ 𝑊 ) |
3 |
|
lcvnbtwn.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝑋 ) |
4 |
|
lcvnbtwn.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑆 ) |
5 |
|
lcvnbtwn.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
6 |
|
lcvnbtwn.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
7 |
|
lcvnbtwn.d |
⊢ ( 𝜑 → 𝑅 𝐶 𝑇 ) |
8 |
|
lcvntr.p |
⊢ ( 𝜑 → 𝑇 𝐶 𝑈 ) |
9 |
1 2 3 4 5 7
|
lcvpss |
⊢ ( 𝜑 → 𝑅 ⊊ 𝑇 ) |
10 |
1 2 3 5 6 8
|
lcvpss |
⊢ ( 𝜑 → 𝑇 ⊊ 𝑈 ) |
11 |
9 10
|
jca |
⊢ ( 𝜑 → ( 𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈 ) ) |
12 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑅 𝐶 𝑈 ) → 𝑊 ∈ 𝑋 ) |
13 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑅 𝐶 𝑈 ) → 𝑅 ∈ 𝑆 ) |
14 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑅 𝐶 𝑈 ) → 𝑈 ∈ 𝑆 ) |
15 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑅 𝐶 𝑈 ) → 𝑇 ∈ 𝑆 ) |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑅 𝐶 𝑈 ) → 𝑅 𝐶 𝑈 ) |
17 |
1 2 12 13 14 15 16
|
lcvnbtwn |
⊢ ( ( 𝜑 ∧ 𝑅 𝐶 𝑈 ) → ¬ ( 𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈 ) ) |
18 |
17
|
ex |
⊢ ( 𝜑 → ( 𝑅 𝐶 𝑈 → ¬ ( 𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈 ) ) ) |
19 |
11 18
|
mt2d |
⊢ ( 𝜑 → ¬ 𝑅 𝐶 𝑈 ) |